ON COOPERATIVE IMAGE DENOISING

Maciej Niedzwiecki and Szymon Gackowski

Faculty of Electronics, Telecommunications and Computer Science, Department of Automatic ControlGdansk University of Technology, Narutowicza 11/12, 80-233 Gdansk, Poland

maciekn@eti.pg.gda.pl, szymon.gackowski@gmail.com

ABSTRACT

In this paper we suggest how several competing image de-noising algorithms, differing in design parameters, or even indesign principles, can be combined together to yield a bet-ter and more reliable denoising algorithm. The proposed fu-sion mechanism allows one to combine practically all kindsof noise reduction tools. It also allows one to account for thedistribution of measurement noise, and in particular – to copewith heavy-tailed disturbances, such as Laplacian noise, orlight-tailed disturbances, such as uniform noise.

Index Terms— Image denoising, Bayesian techniques.

1. INTRODUCTION

Images are often corrupted by noise during their acquisitionand/or transmission. The goal of denoising is to remove thenoise while retaining important image features. Due to itslarge practical importance, image denoising has attracted agreat deal of attention from specialists representing differentdisciplines (microbiology, medicine, radar, astronomy) in thepast 30 years. The typical (and long-standing) solutions to thedenoising problem include: spatially adaptive linear smooth-ing techniques (which utilize local statistics of the image tomake improvement on the filtering performance) [1], [2], non-linear order-statistic filtering [3], [4], and wavelet threshold-ing (shrinkage) procedures [5], [6], to name only the bestknown approaches among the numerous proposed ones.It should be clearly stated that the purpose of this paper is notto describe yet another image smoothing algorithm. Ratherthan proposing a new smoothing paradigm, we will suggesthow several competing smoothers, differing in design param-eters, and/or in design principles, can be combined togetheryielding a better and more reliable smoothing algorithm. Thenew approach, further referred to as cooperative denoising, isa two-dimensional extension of the method proposed in [7]for smoothing of one-dimensional signals.

This work was supported by the Foundation for Polish Science.

2. COOPERATIVE DENOISING

Consider a noisy image G = {g(i, j), i = 1, . . . , I, j =1, . . . , J}:

g(i, j) = f(i, j) + n(i, j) (1)where {f(i, j)} denotes the original (uncorrupted) signalthat should be recovered, and {n(i, j)} denotes measurementnoise – the sequence of independent, identically distributed(i.i.d.) random variables obeying the generalized Gaussianlaw [8]

n ∼ GN (μ, α, β) :

p(n;μ, α, β) =β

2αΓ(1/β)exp

{−

( |n − μ|α

)β}

(2)

where μ is the location parameter, α > 0 is the scaleparameter, β > 0 is the shape parameter, and Γ(x) =∫ ∞0

e−zzx−1dz, for x > 0, denotes the Euler’s gammafunction (extension of the factorial function).Generalized Gaussian is a parametric family of symmetricdistributions that includes normal distribution when β = 2(with mean μ and variance α2/2), and Laplace distributionwhen β = 1 (with mean μ and variance 2α2). When β → ∞,the density (2) converges pointwise to a uniform density on(μ − α, μ + α).We will assume that μ = 0 (zero-mean measurement noise),and that β ≥ 1 is a predetermined (user-defined) shape pa-rameter. We will not assume that the scale parameter α > 0is known – α will be treated as a nussance parameter with as-signed noninformative (improper) prior distribution: π(α) ∝1/α.Our goal is to denoise g(i, j), i.e., to obtain an estimate f(i, j)of f(i, j) which minimizes the mean-squared error (MSE)

E{

[f(i, j) − f(i, j)]2}

(3)

Denote by

fk(i, j) = hk[i, j,G], k = 1, . . . , K (4)

any collection of smoothers corresponding to different de-sign parameters and, possibly, obtained using different designprinciples.

933978-1-4577-0539-7/11/$26.00 ©2011 IEEE ICASSP 2011

Perhaps the simplest examples of smoothers that can be usedfor image denoising are averaging filters (effective in remov-ing noise from homogenous image areas, but blurring edges)

fk(i, j) =

∑(i1,j1)∈Fk(i,j) g(i1, j1)

c[Fk(i, j)](5)

and edge-preserving median filters1

fk(i, j) = med{g(i1, j1), (i1, j1) ∈ Fk(i, j)} (6)

obtained for fitting frames F1(i, j), . . . , FK(i, j), centered at(i, j) and differing in size and/or shape. The quantity c[·] de-notes cardinality (number of elements) of the correspondingset. The size of a fitting frame determines smoothing band-width of the corresponding smoother.We will propose a spatially adaptive mechanism which auto-matically tunes smoothing bandwidth to the local image struc-ture and local signal-to-noise ratio (SNR). Following [7], thecooperative smoother, combining results yielded by all com-peting smoothing algorithms, will be defined in the form

f(i, j) =K∑

k=1

μ◦k(i, j)fk(i, j) , ∀ i, j (7)

where

μ◦k(i, j) =

ϕ◦k(i, j)∑K

k=1 ϕ◦k(i, j)

, k = 1, . . . , K (8)

denote credibility coefficients (related to posterior probabili-ties of different image “patterns”), and the quantities ϕ◦

k(i, j)are evaluated according to

ϕ◦k(i, j) =

⎡⎣ ∑(i1,j1)∈E(i,j)

|ε◦k(i1, j1)|β⎤⎦−M/β

. (9)

whereM = c[E(i, j)] denotes the number of elements of theevaluation frame E(i, j), centered at (i, j). Note that whenβ → ∞ (the uniform noise case), it holds that

ϕ◦k(i, j) −→

[max

(i1,j1)∈E(i,j)|ε◦k(i1, j1)|

]−M

. (10)

The matching errors ε◦k(i, j), used to determine credibility co-efficients, are defined as

ε◦k(i, j) = g(i, j) − f◦k (i, j) (11)

where f◦k denotes holey smoother associated with fk(i, j),

i.e., a smoothing procedure that is identical with (4) except1med{x1, . . . , xn} is defined as x(n+1)/2 for odd values of n, and

(xn/2+xn/2+1)/2 for even values of n, where xi is the ith smallest sampleamong {x1, . . . , xn}.

that it excludes the “central” sample g(i, j) from the set ofmeasurements used for estimation of f(i, j)

f◦k (i, j) = hk[i, j,G◦(i, j)], k = 1, . . . , K (12)

G◦(i, j) = G − {g(i, j)}.A very important property of matching errors ε◦k(i, j) is that,unlike residual errors εk(i, j) = g(i, j) − fk(i, j), they arepointwise independent of measurement noise n(i, j). Ow-ing to this property, they allow one to obtain unbiased (ap-proximately) estimate of the local performance of fk. Holeysmoothers associated with (5) and (6) can be obtained by re-placing the fitting frames Fk(i, j) with their holey counter-parts F ◦

k (i, j) = Fk(i, j) − {(i, j)}.The size M of the evaluation frame E(i, j), independent ofsizes of fitting frames, should be large enough to ensure sta-tistical “stability” of credibility coefficients, but small enoughto guarantee spatial adaptivity of the smoothing scheme. Wehave checked experimentally that a square 9 × 9 window ful-fills both requirements.From the qualitative viewpoint, cooperative smoother can beregarded a Bayesian extension of the one-leave-out cross-validation approach to selection of smoothing bandwidth.Cross-validatory analysis was originated by Stone [9] andfurther developed by many authors [10].

RemarkSome of the quantities involved in computation of credibilitycoefficients μ◦

k(i, j)may take very large or very small values.The following modified expression, mathematically equiva-lent to (8), allows one to avoid numerical problems (such asnumerical overflow) caused by improper scaling

μ◦k(i, j) =

exp{χk(i, j)}∑Kk=1 exp{χk(i, j)}

(13)

where

χk(i, j) = ψk(i, j) − ψmax(i, j)ψk(i, j) = log ϕ◦

k(i, j) = −(M/β) log sk(i, j)ψmax(i, j) = max

1≤k≤Kψk(i, j) .

andsk(i, j) =

∑(i1,j1)∈E(i,j)

|ε◦k(i1, j1)|β .

3. DEMONSTRATION EXAMPLE

To demonstrate the potential of cooperative approach to de-noising, we will examine properties of a very simple schemecombining 4 averaging filters (5) and 4 median filters (6), cor-responding to square 3×3, 5×5, 7×7, and 9×9 fitting frames,respectively. In all tests described below a square 9 × 9 eval-uation frame was used.

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Fig. 1. Performance comparison (MSE, SSIM) of componentavaraging filters (thin lines marked with squares), componentmedian filters (thin lines marked with triangles) and the coop-erative smoother (thick line marked with circles) [Lena imagecorrupted by Gaussian noise (two upper figures) or Laplaciannoise (two lower figures) with standard deviation σn].

Fig. 1 shows comparison of performance of component fil-ters with that of a cooperative scheme. Performance evalu-ation was based on two measures of fit: the mean-squarederror (the smaller the better) and the complex wavelet struc-tural similarity index (the closer to 1 the better). Structuralsimilarity (SSIM) index [11] provides relative scores that aremuch more consonant with our perception of images thanMSE scores. The plots show dependence of average MSEand SSIM scores on standard deviation of an additive Gaus-sian noise (β = 2) and Laplacian noise (β = 1) corruptingthe 512×512 Lena image. All results were obtained for 15realizations of noise. Note that cooperative smoother alwaysworks better than component smoothers. Typical effects ofdenoising are shown in Fig. 2.Fig. 3 shows comparison of the proposed ad hoc smoother(which was not optimized in any way) with the state-of-the-artwavelet thresholding procedures VisuShrink and BayesShrink(for the Daubechies D6 basis). Despite its simplicity, formoderate and large noise intensities the proposed schemeyields better results than procedures based on wavelet thresh-olding. The same conclusion stems from analyzing the av-erage scores obtained for several other 512×512 naturalimages (Barbara, Boat, Peppers, and Zelda) commonly usedfor benchmarking purposes – see Tab. I (all test images wereimported from: http://links.uwaterloo.ca/Repository.html).Finally, Fig. 4 demonstrates spatial adaptivity of the coop-erative smoother. The Boat image was corrupted with spa-tially inhomogeneous Gaussian noise, the standard deviation

Fig. 2. Lena image corrupted by additive Gaussian noise withstandard deviation 25 (left) and the result of its denoising us-ing cooperative smoother (right).

σn image BayesShrink cooperativeMSE SSIM MSE SSIM

Barbara 55.1 0.85 183.7 0.7810 Boat 44.7 0.83 66.2 0.82

Peppers 33.4 0.83 32.7 0.86Zelda 22.7 0.88 20.9 0.90Barbara 134.0 0.73 217.0 0.72

20 Boat 95.1 0.73 99.2 0.76Peppers 69.6 0.77 55.2 0.81Zelda 44.8 0.82 39.4 0.85Barbara 207.9 0.66 257.6 0.67

30 Boat 145.8 0.66 140.0 0.70Peppers 107.7 0.73 84.6 0.76Zelda 64.0 0.79 62.8 0.80Barbara 279.9 0.61 303.1 0.62

40 Boat 191.9 0.62 186.2 0.65Peppers 142.4 0.70 119.0 0.72Zelda 82.6 0.77 90.9 0.75

Table 1. Average scores (MSE, SSIM) obtained for severalclassic natural images using the BayesShrink procedure andcooperative smoother (all results were obtained for 15 real-izations of additive Gaussian noise with standard deviationσn).

of which linearly grew from 5 to 45 along the horizontal im-age axis. Spatially inhomogeneous noise may be encounteredin some medical images [12]. Results of denoising, presentedin Fig. 4, confirm that cooperative smoother copes well withnoise inhomogeneity (MSE=121.3, SSIM=0.74). We notethat, due to global thresholding, all wavelet-based denoisingprocedures fail to work correctly under such circumstances(for BayesShrink the corresponding scores are: MSE=311.7and SSIM=0.58).It should be stressed that the results presented above are justa simple example of benefits that can be gained using cooper-ative approach to denoising.

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0.8

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Fig. 3. Performance comparison (MSE, SSIM) of cooperativesmoother (thick line marked with circles) with two denois-ing procedures based on wavelet thresholding: VisuShrink(thin line marked with diamonds) and BayesShrink (thin linemarked with triangles). [Lena image corrupted by Gaussiannoise (two upper figures) or Laplacian noise (two lower fig-ures) with standard deviation σn].

4. CONCLUSION

We have shown how several competing smoothers, differingin design parameters, or even in design principles, can becombined together yielding a better and more reliable denois-ing algorithm. The proposed scheme allows one to combinepractically all kinds of smoothers. It also allows one to ac-count for the distribution of measurement noise.

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[5] D. Donoho and I. Johnstone, “Adapting to unknownsmoothness via wavelet shrinkage,” American Statisti-

Fig. 4. Results of image denoising in the presence of spatiallyinhomogeneous noise: original Boat image (top left), imagecorrupted by noise (top right), restoration using BayesShrink(bottom left), and restoration using cooperative smoother(bottom right).

cal Assoc., vol. 90, pp. 1200–1224, 1995.[6] G. Chang, B. Yu and M. Vetterli, “Adapting wavelet

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[10] H. Friedl and E. Stampfer, “Cross-validation,” inEncyclopedia of Environmetrics, A.H. El-Shaarawi &W.W. Piegorsch, Eds., vol. 1, pp. 452–460, New York:Wiley, 2002.

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